Greenwood press

64
PERIMETER
A carpenter or artist uses the concept of perimeter
to build frames for pictures and paintings.
Rectangular pen with length l and width w.

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width is 20 meters. If the width is 20 meters, and 
l = 40 − w, then the length is
also 20 meters when the area of the rectangle is a maximum value. Therefore the
ideal rectangular pen based on an existing amount of fencing is a square.
However, if the farmer wants to build a pen that divides different animals,
then the dimensions will have to be reconsidered. For example, suppose there are
chickens and pigs in a pen that is evenly divided by a fence line. In this case, the
dimensions of the most ideal pen would be determined by the equation 
3w
+2l = 80 to account for the added divider. The area of this pen is a =
(40 − 1.5w)w = 40w − 1.5w
2
. A graph of this function shows that the area
attains a maximum value when the width is 
40/3 meters. If the width is 40/3
meters, and 
l = 40 − 1.5w, then the length of the fence should be 20 meters
(40 − 1.5 •
40
3
) when the area of the rectangle is a maximum value. 
In addition to building fences, the concept of perimeter is used in building
race tracks. For example, in track and field, a 400-meter track represents the
perimeter around two congruent semicircles, where the turns and straightaways
are each one hundred meters. Building a track with these dimensions requires
designers and engineers to determine the distance across the track between
straightaways, which represents the diameter of the semicircles. Since the cir-
cumference of a circle is 
π times its diameter, the circumference of a semicircle 
is one-half 
π times its diameter (c =
1
2
πd). Rearranging the variables in the 
equation can show that the diameter, 
d, of a semicircle is d =
2c
π
, where 
c is the 
circumference of the semicircle, which is the 100-meter turn of the track. The 
distance across the infield of a track is
d =
2•100
π
, which is approximately 63.662 
meters.